In this talk, I will
present our results on the study of hybrid lattice models composed
spins and bosons interacting with each other. In particular, I will
address three topics: (i) The possibility of deriving fundamental
limits for the speed of propagation of spin-spin correlations (i.e.
Lieb-Robinson bounds). (ii) The existence of a quantum phase
transition in the Ising universality class that involves spins and
boson on the same footing. (iii) The prospects of finding spin-boson
instances of the Luttinger-liquid universality class. I will also
discuss their connection to ion-trap and superconduncting-cirquit
architectures.